Not sure about DropBox, but AWS has alot of protections around what goes into S3. I have worked with them to address issues that come up in the service, and they couldn't access our data, even when we would have been fine with it. Had to copy it somewhere else for them.
There are protections in both cases. But they are different.
For instance, in the USA, there is no requirement that you have a mechanism to provide information stored at home to law enforcement, even under warrant/court order. If you have one, they can compel you to use it, but if you don't they have no remediation.